Rhombus Calculator
Calculate area, perimeter, and diagonals of a rhombus from side and angle. Enter values for instant results with step-by-step formulas.
Formula
A = s^2 x sin(angle) | d1 = 2s x sin(a/2) | d2 = 2s x cos(a/2)
Where A is the area, s is the side length, angle is any interior angle, d1 is the shorter diagonal, and d2 is the longer diagonal. The area can also be calculated as (d1 x d2) / 2. The perimeter is always 4s since all sides are equal. The diagonals bisect each other at right angles.
Worked Examples
Example 1: Rhombus with Side 10 cm and 60-degree Angle
Problem: Calculate area, perimeter, diagonals, altitude, and inradius for a rhombus with side = 10 cm and acute angle = 60 degrees.
Solution: Area = 10^2 x sin(60) = 100 x 0.8660 = 86.60 cm^2\nPerimeter = 4 x 10 = 40 cm\nDiagonal 1 = 2 x 10 x sin(30) = 10.00 cm\nDiagonal 2 = 2 x 10 x cos(30) = 17.32 cm\nAltitude = 10 x sin(60) = 8.66 cm\nInradius = (10 x sin(60)) / 2 = 4.33 cm\nVerify area via diagonals: (10 x 17.32) / 2 = 86.60 cm^2
Result: Area: 86.60 cm^2 | Perimeter: 40 cm | Diagonals: 10.00, 17.32 cm | Altitude: 8.66 cm
Example 2: Nearly Square Rhombus (80-degree angle)
Problem: A rhombus has sides of 8 cm and an acute angle of 80 degrees. Find all properties.
Solution: Area = 8^2 x sin(80) = 64 x 0.9848 = 63.03 cm^2\nPerimeter = 4 x 8 = 32 cm\nDiagonal 1 = 2 x 8 x sin(40) = 10.28 cm\nDiagonal 2 = 2 x 8 x cos(40) = 12.26 cm\nAltitude = 8 x sin(80) = 7.88 cm\nInradius = (8 x 0.9848) / 2 = 3.94 cm\nSupplementary angle = 180 - 80 = 100 degrees
Result: Area: 63.03 cm^2 | Perimeter: 32 cm | Diagonals: 10.28, 12.26 cm
Frequently Asked Questions
What is a rhombus and what are its defining properties?
A rhombus is a quadrilateral (four-sided polygon) where all four sides are equal in length. It is a special type of parallelogram, which means opposite sides are parallel and opposite angles are equal. The key properties that distinguish a rhombus include four equal sides, opposite angles that are equal, consecutive angles that are supplementary (adding to 180 degrees), diagonals that bisect each other at right angles, and diagonals that bisect the vertex angles. A square is a special case of a rhombus where all angles are 90 degrees. The word rhombus comes from the Greek word rhombos meaning a spinning top or something that spins, referring to the shape that a spinning object traces.
How do I calculate the area of a rhombus?
There are three common methods to calculate the area of a rhombus. The first method uses the side length and an angle: Area = side squared times the sine of any interior angle. The second method uses the two diagonals: Area = (diagonal1 times diagonal2) divided by 2. The third method uses the base and height (altitude): Area = side times altitude, where altitude = side times sine of the angle. All three methods yield the same result. For example, a rhombus with side 10 cm and a 60-degree angle has area = 100 times sin(60) = 100 times 0.866 = 86.60 square centimeters. The diagonal method is often preferred when diagonal measurements are available because it requires only multiplication and division.
How are the diagonals of a rhombus related to its sides and angles?
The diagonals of a rhombus have several important relationships with its sides and angles. The shorter diagonal d1 = 2 times side times sine of half the acute angle, and the longer diagonal d2 = 2 times side times cosine of half the acute angle. The diagonals always bisect each other at right angles (90 degrees), creating four congruent right triangles. Each half-diagonal and the side form a right triangle where the side is the hypotenuse. By the Pythagorean theorem, (d1/2) squared plus (d2/2) squared equals the side squared. The diagonals also bisect the vertex angles, meaning each diagonal splits its vertex angles into two equal parts. The longer diagonal connects the two acute angle vertices, while the shorter diagonal connects the two obtuse angle vertices.
What is the difference between a rhombus and a diamond shape?
In everyday language, the terms rhombus and diamond are often used interchangeably, but there are subtle distinctions. A rhombus is a precise mathematical term for a quadrilateral with four equal sides, defined by its geometric properties regardless of orientation. A diamond shape typically refers to a rhombus oriented with one diagonal vertical (standing on a vertex like a playing card diamond suit). In mathematics, the orientation does not change the shape, so a rotated rhombus is still a rhombus. The diamond shape seen on playing cards, road signs, and baseball fields is geometrically a rhombus or square. In crystallography, the rhombus shape defines the rhombic crystal system, while in common usage, diamond can refer to various pointed shapes that are not strictly rhombi.
What is the inradius of a rhombus and how is it calculated?
The inradius (or apothem) of a rhombus is the radius of the largest circle that fits inside the rhombus, touching all four sides. It is calculated as r = (s times sin(a)) / 2, where s is the side length and a is any interior angle. Alternatively, it equals the area divided by the semi-perimeter: r = Area / (2s). The inradius can also be expressed in terms of the diagonals: r = (d1 times d2) / (4s). For a rhombus with side 10 cm and 60-degree angle, the inradius = (10 times sin(60)) / 2 = (10 times 0.866) / 2 = 4.33 cm. The inscribed circle is tangent to each side at a single point, and its center coincides with the intersection point of the diagonals. Every rhombus has an inscribed circle, making it a tangential polygon.
When does a rhombus become a square?
A rhombus becomes a square when all four interior angles equal 90 degrees. Since a rhombus already has four equal sides, adding the constraint of four right angles satisfies all the requirements of a square. Equivalently, a rhombus is a square when its two diagonals are equal in length, because equal diagonals in a parallelogram produce right angles. In the formulas, when angle a = 90 degrees, sin(90) = 1 so the area becomes s squared (the familiar square area formula), and both diagonals equal s times the square root of 2. A square is simultaneously a rectangle (four right angles), a rhombus (four equal sides), a parallelogram (two pairs of parallel sides), and a regular polygon (all sides and angles equal). It inherits all properties of each of these shape categories.